dd_real atan2(const dd_real &y, const dd_real &x) {
/* Strategy: Instead of using Taylor series to compute
arctan, we instead use Newton's iteration to solve
the equation
sin(z) = y/r or cos(z) = x/r
where r = sqrt(x^2 + y^2).
The iteration is given by
z' = z + (y - sin(z)) / cos(z) (for equation 1)
z' = z - (x - cos(z)) / sin(z) (for equation 2)
Here, x and y are normalized so that x^2 + y^2 = 1.
If |x| > |y|, then first iteration is used since the
denominator is larger. Otherwise, the second is used.
*/
if (x.is_zero()) {
if (y.is_zero()) {
/* Both x and y is zero. */
dd_real::abort("(dd_real::atan2): Both arguments zero.");
return dd_real::_nan;
}
return (y.is_positive()) ? dd_real::_pi2 : -dd_real::_pi2;
} else if (y.is_zero()) {
return (x.is_positive()) ? dd_real(0.0) : dd_real::_pi;
}
if (x == y) {
return (y.is_positive()) ? dd_real::_pi4 : -dd_real::_3pi4;
}
if (x == -y) {
return (y.is_positive()) ? dd_real::_3pi4 : -dd_real::_pi4;
}
dd_real r = sqrt(sqr(x) + sqr(y));
dd_real xx = x / r;
dd_real yy = y / r;
/* Compute double precision approximation to atan. */
dd_real z = std::atan2(to_double(y), to_double(x));
dd_real sin_z, cos_z;
if (xx > yy) {
/* Use Newton iteration 1. z' = z + (y - sin(z)) / cos(z) */
sincos(z, sin_z, cos_z);
z += (yy - sin_z) / cos_z;
} else {
/* Use Newton iteration 2. z' = z - (x - cos(z)) / sin(z) */
sincos(z, sin_z, cos_z);
z -= (xx - cos_z) / sin_z;
}
return z;
}
dd_real acos(const dd_real &a) {
dd_real abs_a = abs(a);
if (abs_a > 1.0) {
dd_real::abort("(dd_real::acos): Argument out of domain.");
return dd_real::_nan;
}
if (abs_a.is_one()) {
return (a.is_positive()) ? dd_real(0.0) : dd_real::_pi;
}
return atan2(sqrt(1.0 - sqr(a)), a);
}
dd_real asin(const dd_real &a) {
dd_real abs_a = abs(a);
if (abs_a > 1.0) {
dd_real::abort("(dd_real::asin): Argument out of domain.");
return dd_real::_nan;
}
if (abs_a.is_one()) {
return (a.is_positive()) ? dd_real::_pi2 : -dd_real::_pi2;
}
return atan2(a, sqrt(1.0 - sqr(a)));
}